Table of Contents Foreword iii Table of Contents v Notation xi 1 Partial Differential Equations and Their Classification Into Types 1 1.1 Examples 1 1.2 Classification of Second-Order Equations into Types 4 1.3 Type Classification for Systems of First Order 6 1.4 Characteristic Properties of the Different Types 7 2 The Potential Equation 12 2.1 Posing the Problem 12 2.2 Singularity Function 14 2.3 The Mean Value Property and Maximum Principle 17 2.4 Continuous Dependence on the Boundary Data 23 3 The Poisson Equation 27 3.1 Posing the Problem 27 3.2 Representation of the Solution by the Green Function 28 3.3 The Green Function for the Ball 34 3.4 The Neumann Boundary Value Problem 35 3.5 The Integral Equation Method 36 4 Difference Methods for the Poisson Equation 38 4.1 Introduction: The One-Dimensional Case 38 4.2 The Five-Point Formula 40 4.3 M-matrices, Matrix Norms, Positive Definite Matrices 44 4.4 Properties of the Matrix Lh 53 4.5 Convergence 59 4.6 Discretisations of Higher Order 62 4.7 The Discretisation of the Neumann Boundary Value Problem 65 4.7.1 One-sided Difference for θu/θn 65 4.7.2 Symmetric Difference for θu/θn 70 4.7.3 Symmetric Difference for θu/θn on an Offset Grid 71 4.7.4 Proof of the Stability Theorem 7 72 4.8 Discretisation in an Arbitrary Domain 78 4.8.1 Shortley-Weller Approximation 78 4.8.2 Interpolation at Points near the Boundary 83 5 General Boundary Value Problems 85 5.1 Dirichlet Boundary Value Problems for Linear Differential Equations 85 5.1.1 pesing the Problem 85 5.1.2 Maximum Principle 86 5.1.3 Uniqueness of the Solution and Continuous Dependence 87 5.1.4 Difference Methods for the General Differential Equation of Second Order 90 5.1.5 Green’s Function 95 5.2 General Boundary Conditions 95 5.2.1 Formulating the Boundary Value Problem 95 5.2.2 Difference Methods for General Boundary Conditions 98 5.3 Boundary Problems of Higher Order 103 5.3.1 The Biharmonic Differential Equation 103 5.3.2 General Linear Differential Equations of Order 2m 104 5.3.3 Discretisation of the Biharmonic Differential Equation 105 6 Tools from Functional Analysis 110 6.1 Banach Spaces and Hilbert Spaces 110 6.1.1 Normed Space 110 6.1.2 Operators 111 6.1.3 Banach Spaces 112 6.1.4 Hilbert Spaces 114 6.2 Sobolev Space 115 6.2.1 L2(Ω) 115 6.2.2 Hk(Ω) and Hk0(Ω) 116 6.2.3 Fourier Transformation and Hk(IRn) 119 6.2.4 H8(Ω) for Real s≥0 122 6.2.5 Trace and Extension Theorems 123 6.3 Dual Spaces 130 6.3.1 Dual Space of a Normed Space 130 6.3.2 Adjoint Operators 132 6.3.3 Scales of Hilbert Spaces 133 6.4 Compact Operators 135 6.5 Bilinear Forms 137 7 Variational Formulation 144 7.1 Historical Remarks 144 7.2 Equations with Homogeneous Dirichlet Boundary Conditions 145 7.3 Inhomogeneous Dirichlet Boundary Conditions 150 7.4 Natural Boundary Conditions 152 8 The Method of Finite Elements 161 8.1 The Ritz-Galerkin Method 161 8.2 Error Estimates 167 8.3 Finite Elements 171 8.3.1 Introduction: Linear Elements forΩ= (a,b) 171 8.3.2 Linear Elements forΩIR2 174 8.3.3 Bilinear Elements forΩIR2 178 8.3.4 Quadratic Elements forΩIR2 180 8.3.5 Elements for nΩIR2 182 8.3.6 Handling of Side Conditions 182 8.4 Error Estimates for Finite Element Methods 185 8.4.1 H1-Estimates for Linear Elements 185 8.4.2 L2 and H8 Estimates for Linear Elements 190 8.5 Generalisations 193 8.5.1 Error Estimates for Other Elements 193 8.5.2 Finite Elements for Equations of Higher Order 194 8.5.2.1 Introduction: The One-Dimensional Biharmonic Equation 194 8.5.2.2 The Tw。Dimensional Case 195 8.5.2.3 Estimating Errors 196 8.6 Finite Elements for Non-Polygonal Regions 196 8 7 Additional Remarks 199 8.7.l Non-Conformal Elements 199 8.7.2 The Trefftz Method 200 8.7.3 Finitt Element Methods for Singular Solutions 201 8.7.4 Adaptive Triangulation 201 8.7.5 Hierarchical Bases 202 8.7.6 Superconvergence 202 8.8 Properties of the Stiffness Matrix 203 9 Regularity 208 9.1 Solutions of the Boundary Value Problem in Hs(Ω),s>m 208 9.1.1 The Regularity Problem 208 9.1.2 Regularity Theorems for Ω= IRn 210 9.1.3 Regularity Theorems forΩ= IRn + 215 9.1.4 Regularity Theorems for General Ω IRn 219 9.1.5 Regularity for Convex Domains and Domains with Corners 223 9.1.6 Regularity in the Interior 226 9.2 Regularity Properties of Difference Equations 226 9.2.1 Discrete H1-Regularity 226 9.2.2 Consistency 232 9.2.3 Optimal Error Estimates 238 9.2.4 H2h-Regularity 240 10 Special Differential Equations 244 10.1 Differential Equations with Discontinuous Coefficients 244 10.1.1 Formulation 244 10.1.2 Discretisation 246 10.2 A Singular Perturbation Problem 247 10.2.1 The Convection-Diffusion Equation 247 10.2.2 Stable Difference Schemes 249 10.2.3 Finite Elements 251 11 Eigenvalue Problems 253 11.1 Formulation of Eigenvalue Problems 253 11.2 Finite Element Discretisation 254 11.2.1 Discretisation 254 11.2.2 Qualitative Convergence Results 256 11.2.3 Quantitative Convergence Results 260 11.2.4 Complementary Problems 264 11.3 Discretisation by Difference Methods 267 12 Stokes Equations 275 12.1 Systems of Elliptic Differential Equations 12.2 Variational Formulation 278 12.2.1 Weak Formulation of the StokesEquations 278 12 2.2 Saddlepoint Problems 279 12.2.3 Existence and Uniqueness of the Solution of a Saddlepoint Problem 282 12.2.4 Solvability and Regularity of the Stokes Problem 285 12.2.5 A V0-elliptic Variational Formulation of the Stokes Problem 289 12.3 Mixed Finite-Element Method for the StokesProblem 290 12.3.1 Finite-Element Discretisation of a Saddlepoint Problem 290 12.3.2 Stability Conditions 291 12 3.3 Stable Finite-Element Spaces for the StokesProblem 293 12.3.3.1 Stability Criterion 293 12.3.3.2 Finite-Element Discretisations with the Bubble Function 294 12.3.3.3 Stable Discretisations with Linear Elements in Vh 296 12.3.3.4 Error Estimates 297 Bibliography 300 Index 307
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